Generalizations of the geometric phase have been studied for cyclic states, i.e. states that return to the same ray in state space after some time. Using left-eigenstates instead of bra's, a generalization of Berry's phase for cyclic eigenstates of a non-degenerate non-Hermitian Hamiltonian was presented in . A geometric model for such cases was introduced in , which uses a parameter space augmented by information from the energy levels. We also mention that other generalizations for Hermitian systems have been found, e.g. for degenerate Hamiltonians  or for any cyclic state in a Hilbert space [1]}.
